A couple. Two people one thinks should date, or likes that they do/did. Usually combines the two names to make a ship name, i.e.:John+Karen=Jaren. Usually the boy's name goes first, however it is used. It can combine the last names to such as peazer+payne=payzer. A rare form of a ship name is one name inside another, such as louis+eleanor=elounor(i don't ship it). A ship is the couple. To ship something is to suppourt the ship. Shipping is when you create ships. Ships are most commonly used in fandoms or with celebrities, but are also used for people you know. Also commonly used for fictional characters.

"Do you ship Joe and Stacey?"
"Yes! Their ship name should be Stoe. Or Jacey!"
"Oh, they are such a cute couple."

Weirdly, everyone seems to see this as an abbreviation of "relationship", and use it to profess a desire for two people to couple or "pair up". Whoever started popular use of this his having a giggle on all of you, as the proper derivation of it is simply

A term when one person likes the idea of two people in a relationship. This term can be used on an actual couple or any two people someone may find good together. When someone "ships" two people they will most likely come up with a ship name for them. This is when you take the first names of the two people and mash them together (typically the male's name going first, although two males or two females can also be shipped).

Person 1: Omg! They Kian and Andrea are so cute! I ship them!
Person 2: Me too! What is their ship name?
Person 1: Kiandrea!